The radioactive contamination of a BaF$_2$ scintillation crystal with mass of 1.714 kg was measured over 101 hours in the low-background DAMA/R&D set-up deep underground (3600 m w.e.) at the Gran Sasso National Laboratories of INFN (LNGS, Italy). The half-life of $^{212}$Po (present in the crystal scintillator due to contamination by radium) was measured as $T_{1/2}(^{212}$Po) = 298.8$pm$0.8(stat.)$pm$1.4(syst.) ns by analysis of the events pulse profiles. The $^{222}$Rn nuclide is known as 100% decaying via emission of $alpha$ particle with $T_{1/2}$ = 3.82 d; however, its $beta$ decay is also energetically allowed with $Q_beta = 24pm21$ keV. Search for decay chains of events with specific pulse shapes characteristic for $alpha$ or for $beta/gamma$ signals and with known energies and time differences allowed us to set, for the first time, the limit on the branching ratio of $^{222}$Rn relatively to $beta$ decay as $B_beta < 0.13$% at 90% C.L. (equivalent to limit on partial half-life $T_{1/2}^beta > 8.0$ y). Half-life limits of $^{212}$Pb, $^{222}$Rn and $^{226}$Ra relatively to $2beta$ decays are also improved in comparison with the earlier results.
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